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2 Advances n Water Resources 34 (2011) Contents lsts avalable at ScVerse ScenceDrect Advances n Water Resources journal homepage: MAST-2D dffusve model for flood predcton on domans wth trangular Delaunay unstructured meshes C. Arcò a,, M. Snagra a, L. Begnudell b, T. Tuccarell a a Dpartmento d Ingegnera Cvle, Ambentale ed Aerospazale, Unverstà d Palermo, Vale delle Scenze Ed. 8, Palermo, Italy b Dpartmento d Ingegnera Cvle ed Ambentale, Unverstà degl Stud d Trento, Va Mesano 77, Trento, Italy artcle nfo abstract Artcle hstory: Receved 11 Aprl 2011 Receved n revsed form 8 August 2011 Accepted 12 August 2011 Avalable onlne 5 September 2011 Keywords: Shallow waters Fnte element method Dffusve model Unstructured meshes Delaunay trangulatons Analytcal soluton A new methodology for the soluton of the 2D dffusve shallow water equatons over Delaunay unstructured trangular meshes s presented. Before developng the new algorthm, the followng queston s addressed: t s worth developng and usng a smplfed shallow water model, when well establshed algorthms for the soluton of the complete one do exst? The governng Partal Dfferental Equatons are dscretzed usng a procedure smlar to the lnear conformng Fnte Element Galerkn scheme, wth a dfferent flux formulaton and a specal flux treatment that requres Delaunay trangulaton but entre soluton monotoncty. A smple mesh adjustment s suggested, that attans the Delaunay condton for all the trangle sdes wthout changng the orgnal nodes locaton and also mantans the nternal boundares. The orgnal governng system s solved applyng a fractonal tme step procedure, that solves consecutvely a convectve predcton system and a dffusve correcton system. The non lnear components of the problem are concentrated n the predcton step, whle the correcton step leads to the soluton of a lnear system of the order of the number of computatonal cells. A sem-analytcal procedure s appled for the soluton of the predcton step. The dscretzed formulaton of the governng equatons allows to handle also wettng and dryng processes wthout any addtonal specfc treatment. Local energy dsspatons, manly the effect of vertcal walls and hydraulc jumps, can be easly ncluded n the model. Several numercal experments have been carred out n order to test (1) the stablty of the proposed model wth regard to the sze of the Courant number and to the mesh rregularty, (2) ts computatonal performance, (3) the convergence order by means of mesh refnement. The model results are also compared wth the results obtaned by a fully dynamc model. Fnally, the applcaton to a real feld case wth a Ventur channel s presented. Ó 2011 Elsever Ltd. All rghts reserved. 1. Introducton The 2D Sant-Venant (SV) [45], or shallow water (SW) equatons, are extensvely used for hydrodynamc smulatons n rvers, lakes, estuares and floodplans. Wthn the framework of the orgnal SV equatons, the resultng mathematcal model may be classfed as dynamc, gravty, dffuson and knematc wave model, correspondng to dfferent forms of the momentum equaton, respectvely [19,58,17]. Dynamc wave model retans all the terms of the momentum equaton, whereas gravty wave model neglects the effects of bed slope and vscous energy loss and descrbes flows domnated by nerta. As a matter of fact, the acceleraton terms n the SV Correspondng author. E-mal addresses: arco@dra.unpa.t (C. Arcò), snagra@dra.unpa.t (M. Snagra), lorenzo.begnudell@untn.t (L. Begnudell), tuccar@dra.unpa.t (T. Tuccarell). equatons can be neglected n most practcal applcatons of flood routng n natural channels. The system s thus reduced to a sngle parabolc equaton known as the dffusve wave model. If the water depth gradent term s further omtted, the knematc wave equaton s acqured. The crtera for demarcatng knematc and dffusve waves have been fully dscussed [39 41,47 49,55]. The knematc model can be easly solved n the case of steep slope or n ntally dry areas, where soluton of some dffusve models s plagued by nstablty problems. On the other hand, the knematc model s not able to compute backwater effects and provdes physcally nconsstent results when local mnma are present n the topographc surface. The choce of the model to be used for the SW equatons soluton (fully dynamc, dffusve wave, knematc wave, 1D or 2D, etc.) depends also on the avalable nput data and on the capablty of generatng the requred hydraulc nformaton n an approprate format and detal level [50]. These nformaton are, for nstance, the topography, the hydraulc propertes of the rver reaches and /$ - see front matter Ó 2011 Elsever Ltd. All rghts reserved. do: /j.advwatres

3 1428 C. Arcò et al. / Advances n Water Resources 34 (2011) the nundaton zone, the shape of the nput flood hydrographs. For example, the soluton of the fully dynamc equatons s very senstve to the topographc error, as wll be shown n the next secton, and smplfed models such as the knematc wave and the unform formulae do not enable to represent all hydraulc processes durng flood events. The dffusve wave used heren s robust wth respect to the nput data approxmatons, but provdes a hgher order accuracy wth respect to the knematc wave and the unform formulae. Wth excepton of catastrophc events lke dam breaks, flow over nundated plans s often a slow and shallow phenomenon where local free surface slopes are very small. When flood events occur, water s no more contaned n the man channel and splls onto the adjacent floodplans. The physcal process becomes very complex to smulate and s no longer satsfactorly represented by a 1D scheme. 2D numercal technques capable of smulatng floodplan nundatons have been extensvely developed n the last years [24]. Numercal technques, lke fnte volumes (FV) and fnte elements (FE), as well as more conceptually approaches, lke storage cell solutons, have been mplemented for the soluton of these type of problems. In FV and FE methods the doman s parttoned nto cells and the governng equatons, wrtten n conservatve form, are ntegrated n each cell. These methods can be appled to both structured and unstructured meshes. Durng the last two decades, FV Godunov-type schemes have become popular n seekng the numercal soluton of the SW equatons. In such schemes a local Remann problem s solved at every cell nterface. Most of these schemes [2,4,34,54,32] have the capablty of shock capturng wth hgh accuracy level, but perform well for partcular types of flows, for example dscontnuous flows over flat topographes and fal n cases of rregular and varable topography or n the appearance of dry areas. Bermudez and Vazquez [14] and Vazquez-Cendon [57] used an upwnd dscretzaton of the source term over rregular topography and ntroduced the concept of C-property: a numercal scheme s regarded as well balanced or satsfyng the C-property f t preserves steady-state at rest (stagnant condtons). Upwndng of the source terms s computatonally expensve because the source terms have to be projected on a base of the egenvectors. LeVeque [31] ntroduced a Remann problems nsde a cell for balancng the source terms and the flux gradents. The method preserve the C-property and the quas steady-state condtons, but cannot be drectly appled to unstructured grds. Alcrudo and Benkhaldoun [3] used a topography dscretzaton such that a sudden change n the topography occurs at the nterface of two cells and solve a Remann problem at the nterface wth a sudden change n the bed elevaton. Ths approach leads to several cases of Remann fan and results are computatonally very expensve. Zhou et al. [61] ntroduced the surface gradent method, usng the water surface elevaton to calculate the water depth at cell nterfaces. The proposed method mantans the C-property and performs well over varable topography wthout extra efforts for balancng the source terms and the flux gradents. However the C-property does not hold for unstructured grds. Several FE approaches have been developed for the SW equatons over the past two decades; see, for example, [27,28, 33,52,62]. Much of ths effort has nvolved dervng methods whch are stable and non-oscllatory under hghly varyng flow regmes. In recent years, FE methods based on dscretzng the prmtve form of the SW equatons usng dscontnuous approxmatng spaces have also been studed [1,2,15,18]. Ths dscontnuous Galerkn (DG) approach has several appealng features; n partcular, the ablty to ncorporate upwndng and post-processng stablty nto the soluton of hghly advectve flows. Ths approach generalzes and extends the Godunov methods: the hgher-order polynomals are naturally bult nto the method and they are defned through the varatonal equaton, nstead of computng these hgher-order terms by means of ad hoc post processng procedures; dffusve terms are ncorporated n the method, whle, on the opposte, Godunov schemes do not provde any mechansm for dealng wth second-order dervatves. The DG methods allows for the use of non-conformng grds, that have very useful feature n dealng wth complcated geometres. Moreover, the DG method s locally conservatve, that s, the prmtve contnuty equaton relatng the change n water elevaton to water flux s satsfed n a weak sense element by element. The man drawback of DG methods compared to contnuous Galerkn methods s ther addtonal cost. In a DG method, the degrees of freedom of the soluton are assocated wth elements rather than nodal values, and n unstructured Fnte Element meshes, there can be substantally more elements than nodes. One of the man dffculty n the soluton of the SW equatons s the flow computaton over ntally dry areas. If no specal attenton s pad, standard numercal procedure may fal near dry/wet front, producng unphyscal oscllatons and negatve water depths. Durng the last 30, 40 years hydrodynamc models have been equpped wth Wettng Dryng (WD) algorthms [26]. Maybe the most natural WD approach would be to track the WD nterface n tme, movng the boundary nodes and deformng accordngly the computatonal mesh, but a sgnfcant computatonal cost has to be pad for the mesh deformaton. For the above mentoned reasons, most of the avalable WD methods have been developed for fxed mesh. The fxed mesh approach can be dvded n two man categores. In the frst one, ether nodes or entre elements are deactvated when become dry and excluded from the computatonal doman. However, ths ncluson/excluson of elements may volate the mass and momentum conservaton and nfrnge the numercal stablty. In order to descrbe WD nterface that do not match the element nterface, some authors [21] ntroduced transton elements those where some, but not all nodes, are dry requrng specal treatment. Commonly the transton elements are explctly detected and ther pressure gradent term s neglected. Such dscontnuous swtches (as cancellng the pressure gradents) make these methods hghly non lnear and may ntroduce oscllatons and numercal nstabltes. Another class of fxed grd WD technques s the artfcal porosty one [21], where the bed s assumed to be porous and non zero water fluxes are allowed for negatve depths. The man advantage of ths procedure s that the artfcal pressure gradent problem s crcumvented. For more detals about the WD technques, see [21,26]. Most of the recently proposed floodplans nundaton models couple a 1D and a 2D model [35,16,11,23]. In Cunge-type storage models, cells correspond to dstnct flood compartments and geometrc relatonshps based on water levels are constructed to determne the storage for each flood basn. Wth the developments n GIS software these relatonshps are automatcally generated from hgh resoluton Dgtal Elevaton Models (DEMs). The abundance of topographc data processed, stored and manpulated wthn GIS systems has recently led to a fuson of the storage cell concept wth raster data format. Such schemes normally use 1D models for man channel flow routng and dscretze the floodplans by structured Cartesan (or raster) grd. Each floodplan pxel n the grd s treated as an ndvdual storage cell wth nter-cell fluxes treated usng unform flow formulae [11]. The nteracton between the man channel and the floodplans s modelled by wer type equatons. Compared to fully explct Fnte Elements, fnte dfferences and fnte volumes models, raster-based models have an advantage

4 C. Arcò et al. / Advances n Water Resources 34 (2011) n terms of easy formulaton, though questons reman about ther smple representaton of the flow process [60]. In the present work, a numercal methodology for the soluton of dffusve shallow water problem s presented. The governng Partal Dfferental Equatons (PDEs) are dscretzed over unstructured trangulatons usng a procedure smlar to the lnear conformng P1 FE Galerkn scheme but wth a dfferent flux formulaton. The methodology follows a fractonal tme step approach, solvng sequentally a predcton and a correcton problem. The non lnear components of the orgnal PDEs problem are concentrated n the predcton step, whle the correcton step leads to the soluton of a lnear system, of the order equal to the number of computatonal cells. Numercal fluxes dscretzaton s the same n both predcton and correcton steps. A proof s gven to show that the method s both locally and globally mass conservatve. The predcton step s solved applyng the MArchng n Space and Tme (MAST) methodology, recently proposed for the soluton of advecton domnated problems [10,5], of the fully dynamc SW equatons [6,8], as well as of transport problems n saturated porous meda wth varable densty [7]. MAST pecularty s to solve at each tme step one computatonal cell after the others accordng to a gven order, such that the mean enterng flux s known before the cell soluton. Ths provdes an uncondtonal stablty wth respect to the tme step sze, also for Courant (CFL) number much greater than 1. The requrement for the applcaton of the MAST methodology s the exstence of an exact or approxmated scalar potental for the flow feld. In the present physcal problem, an exact scalar potental of the flow feld exsts and t s the pezometrc head. At the begnnng of each tme step, computatonal cells are ordered accordng the ther pezometrc value. MAST solves a sequence of Ordnary Dfferental Equatons (ODEs), one for each computatonal cell, from the hghest to the lowest potental value. The present paper s organzed as follows: the choce of the dffusve model wth respect to the fully dynamc one s frst motvated n Secton 2 and the governng equatons of the dffusve SW equatons are presented n Secton 3, as well as the appled fractonal tme step procedure. The spatal dscretzaton of the orgnal governng equaton system and the MAST scheme are presented n Secton 4, wth the numercal flux formulaton n a Delaunay trangulaton. In Secton 5, a smple procedure to obtan a Delaunay mesh from a gven set of nodes, also ncludng a subset of fxed edges, s provded. Detals of the sem-analytcal procedure for the soluton of the predcton step are gven n Secton 6. Handlng wettng and dryng processes s dscussed n the same secton. The ncluson of the effect of vertcal walls and hydraulc jumps n the proposed model s descrbed n Secton 7. In Secton 8, several numercal experments have been carred out n order to test the effcency and stablty of the proposed model wth regard to the sze of the CFL number, the computatonal performance, as well as the convergence order by mesh refnement, whch s close to 2. Numercal results n the case of flow n a rectangular channel are compared wth the correspondng ones obtaned by other lterature models. The floodng from a composte trapezodal cross secton channel n steady-state condtons s studed, as well as the applcaton of a real feld case wth a Ventur channel. Results of these two last tests are compared wth the correspondng ones computed by the fully dynamc model proposed n [12,13]. 2. Why t can be worth usng a dffusve model nstead of a fully dynamc one? Before presentng the new algorthm for the soluton of the 2D dffusve shallow water problem (DSW), we provde some most mportant motvatons to prefer the dffusve model nstead of the fully dynamc one (FSW). The motvatons can be summarzed as follows: (1) The numercal soluton of the dffusve model can be computed more quckly, for gven mesh sze and smulated tme, (2) only one boundary condton (b.c.) s requred at each boundary pont, where the approprate number of b.c. n the fully dynamc case can be zero, two or three dependng on the local Froude number, and (3) most mportant, the senstvty of the computed water depth to the topographc error s much hgher n the FSW model than n the DSW one. Motvaton (1) s based on the possblty to merge the momentum equatons n the contnuty equaton, n order to get a sngle hgher order equaton n only one unknown (nstead of three unknowns as for the FSW model) and on the exstence of an exact potental. The exact potental and the rrotatonalty of the flow feld allow the applcaton of the MAST procedure, wth tme steps leadng to CFL numbers much larger than one. On the other hand, we have already seen n the ntroducton that mportant advances n the soluton of the FSW model have recently attaned a very good computatonal effcency and have made ths motvaton less compellng than the others. Motvaton (2) s based not only on computatonal advantages, but also on data lmtaton. For example, the avalablty of data regardng supercrtcal flows enterng the upstream doman boundary s often mssng and n the FSW soluton the normal (.e. unform) flow condton s usually adopted to relate water depth to dscharge. Motvaton (3) s the most mportant one. Gunot and Cappallaere [22] have recently analyzed the senstvty of a FSW 2D model wth respect to the parameter errors, where parameters are the topographc elevaton, the Mannng coeffcent and the bed slope. They have shown that, n the very smple case of frctonless, horzontal bed wth unform steady-state flow, the senstvty can be computed as the soluton of a Laplace equaton, where the source term s proportonal to the quantty: a ¼ð1 F 2 r Þ 1=2 where F r s the Froude number. It s well-known that the dffusve model can be thought as a fully dynamc one where the gravty force goes to nfnty. Ths s equvalent to say that the Froude number goes to zero and, n Eq. (1), the quantty a attans ts mnmum possble sze. The same conclusons can be obtaned for the 1D case by observng the behavour of the water depth when a topographc change s gven for constant energy value. See n Fg. 1 the E(h) curve, where h s the water depth and E s the energy per unt weght and constant dscharge, relatve to the bed level (E = h + V 2 /2g, where V s the mean flow velocty and g the gravty acceleraton). The straght lne s the potental component of the energy, that s the same water depth h. When a topographc Dz change locally occurs, E decreases to E Dz. In the dffusve model, h decreases also to E Dz and the pezometrc level remans constant. In the fully dynamc model a larger varaton Dh occurs (see Fg. 1), because the water depth reducton has also to balance the velocty and the correspondng knetc energy ncrement. If the ntal water depth s close to the crtcal value, the water depth senstvty n the fully dynamc model approaches nfnty, as also suggested by Eq. (1) when the Froude number s close to one. The senstvty of the water depth wth respect to the topographc error n the complete model strongly overcomes the same senstvty n the dffusve model only when the Froude number approaches one. On the other hand, the dfference between the results of the two models s sgnfcant only for the same range of Froude number. If the flow s strongly supercrtcal, the water ð1þ

5 1430 C. Arcò et al. / Advances n Water Resources 34 (2011) respectvely, H D and h D are the assgned Drchlet values for H and h, g N s the assgned Neumann flux, q(x,t) s the boundary flow rate vector, n s the unt outward normal to the boundary, x =(x,y) and the subscrpt 0 marks the ntal state n the doman. Eqs. (2b) and (2c) can be merged n Eq. (2a), to get the p n h 5=3 p n h 5=3 ¼ p In the proposed procedure, numercal soluton of Eq. (4) n the H unknown s attaned by means of a tme-splttng approach, solvng consecutvely a predcton and a correcton system. Assume a general system of balance þ r FðUÞ ð5þ where U s the vector of the unknown varables, F(U) s the flux vector and B(U) s a source term. Applyng a fractonal tme step procedure, we set: ð4þ Fg. 1. Varatons of water depth n the dffusve model and complete model resultng from local elevaton change. depth s lkely to resemble ts normal value; f the flow s strongly subcrtcal, the nertal terms are neglgble n the momentum equaton. The choce of the model type stll remans subjectve and casedependent. In the case of dam-break flows or waves wth a length much shorter than the doman extenson, nertal terms prevals n the momentum equaton and the use of a complete model s compulsory. In all the other cases t s our opnon that dffusve models provde more robust and relable solutons, as t wll be shown to happen n the last two numercal tests, manly because of the smaller senstvty to the nput data error and uncertanty. 3. Governng equatons system and the fractonal tme step methodology The 2D dffusve form of the shallow water equatons can be wrtten as a system of three frst order ¼ p ð2aþ p r x H þ n2 u ffffffffffffffffffffffffffffffff u 2 þ v 2 ¼ 0 ð2bþ h 4=3 p v ffffffffffffffffffffffffffffffff r y H þ n2 u 2 þ v 2 ¼ 0 ð2cþ h 4=3 where h s the water depth, H = z + h s the pezometrc head (z s the topographc head), u and v are the x and y velocty components, n s the Mannng coeffcent, r x(y) H s the component of the spatal gradent of the pezometrc head n x(y) drecton, p represents a source term (e.g., ran ntensty). Eq. (2a) s the mass conservaton equaton and Eqs. (2b) and (2c) are the momentum equatons n x and y drectons. Intal and boundary condtons have to be specfed to make problem (2) well posed. Boundary condtons may be of Drchlet (prescrbed pezometrc head or water depth) or Neumann (prescrbed flux) type. If X s the spatal doman where problem (2) s defned, ntal and boundary condton can be wrtten as: hðx; tþ ¼h D ðx; tþ or Hðx; tþ ¼H D ðx; tþ; x 2 C D qðx; tþn ¼ g N ðx; tþ; x 2 C N hðx; 0Þ ¼h 0 or Hðx; 0Þ ¼H 0 ; x 2 X where C = C D [ C N s the boundary of X, C D and C N are the portons of C where Drchlet and Neumann boundary condtons hold ð3þ FðUÞ ¼F p ðuþþðfðuþ F p ðuþþ ð6aþ BðUÞ ¼B p ðuþþðbðuþ B p ðuþþ ð6bþ where F p (U) and B p (U) are respectvely a sutable numercal flux and source term, further defned. After ntegraton n tme, system (5) can be splt n the two followng ones: U kþ1=2 U k þ r U kþ1 U kþ1=2 þ r Z Dt 0 Z Dt 0 F p dt ¼ Z Dt 0 B p dt Fdt r F p Dt ¼ Z Dt 0 Bdt B p Dt ð7aþ ð7bþ where F p and B p are the mean numercal flux and source terms computed along the predcton step, U k+1/2 andu k+1 are the unknown varables computed respectvely at the end of the predcton and the correcton phases. Integrals F p Dt and B p Dt wll be estmated a posteror after the soluton of the predcton problem, accordng to the procedure explaned n the next secton. We call systems (7a) and (7b) predcton and correcton systems respectvely. Observe that summng systems (7a) and (7b), the ntegral of the orgnal system (5) s formally obtaned. The dfference between U k+1 and U k+1/2 n Eq. (7b) s close to zero as far as the dfference between the predcted and mean n tme values of the fluxes and source terms s small. The advantage of usng formulatons (7) nstead of (5) s that, wth a sutable choce of the predcton terms F p (U) and B p (U), each of the two systems (7a) and (7b) can be much easer to solve than the orgnal system (5). In the present case we have: U ¼ H ð8aþ F ¼ p h5=3 n ffffffffffffffffffffff rh jrhj ð8bþ B ¼ p ð8cþ We set: F p ¼ q h5=3 ffffffffffffffffffffffff ðrhþ k ð9aþ n jrhj k B p ¼ B ð9bþ where ndex k marks the begnnng of the tme step (tme level t k ). Observe that the flux formulaton of the predcton step dffers from the orgnal one (Eq. (4)) n the tme level of the gradents of H. In the predcton step, spatal gradents of the pezometrc head are assumed constant n tme and equal to the values computed at the end of the prevous tme step. The predcton equaton to be solved along the gven tme step s:

6 n h 5=3 ffffffffffffffffffffffff jrh k 1 0 C n h 5=3 ffffffffffffffffffffffff jrh k j C. Arcò et al. / Advances n Water Resources 34 (2011) C A ¼ p ð10þ The predcton problem s solved n ts ntegral form, as shown n the followng, whle the correcton problem s solved n ts dfferental lnearzed form: 0 5= @ n ffffffffffffffffffffffff jrh k #ÞC B ðh km Þ qffffffffffffffffffffffff A n jrh j ð11þ P e where g ¼ H H kþ1=2 ð12aþ h km ¼ hk þ h kþ1=2 2 ð12bþ # ¼ H k H kþ1=2 ð12cþ wth ntal condton g = 0. Index k + 1/2 marks the soluton of the predcton Eq. (10). After some smple manpulatons, t can be shown that the quas-lnear dfferental form of the predcton problem s knematc, wth only one characterstc lne passng through each (x, t) pont. The predcton PDE system s equvalent to a sngle non-lnear convecton equaton, functon of the gradent of the pezometrc head at tme level t k, whle correcton system has the functonal characterstcs of a pure dffusve process. For these reasons we call the predcton and the correcton systems respectvely convectve predcton system and dffusve correcton system. The convectve predcton problem has to be solved by gvng the known dscharge as boundary condton to the upstream nodes. The dffusve correcton system s solved by settng to zero dffusve flux n the upstream boundary nodes and by gvng to the downstream nodes the proper boundary condton requred to satsfy the boundary condtons of the orgnal problem (2). For example, f the downstream water level s known and equal to H, the correcton g n the downstream boundary wll be set equal to: g = H H k+1/2. 4. The MAST procedure The spatal dscretzaton of the governng equatons s carred out on a generally unstructured trangular mesh that satsfes the Delaunay property. A Delaunay trangulaton n R 2 s defned by the condton that all the nodes n the mesh are not nteror to the crcles defned by the three nodes of each trangle. Let X R 2 be a bounded doman, X h a polygonal approxmaton of X and T h an unstructured Delaunay-type trangulaton of X h. The trangulaton T h s called basc mesh and the trangle k T 2 T h s called prmary element. Let P h ={P, =1,...,N} be the set of all vertces (nodes) of all k T 2 T h and N the total number of nodes. The dual mesh E h ={e, =1,...,N} s constructed over the basc mesh. The dual fnte volume e assocated wth the vertex P s the closed polygon gven by the unon of sub-trangles resultng from the subdvson of each trangle of T h connected to node P by means of ts axes (see Fg. 2). In the followng of the paper the dual volumes e are called also cells. The sub-trangles are called secondary elements and are ndcated as e II. Cells e satsfy: X ¼[e ð13þ The dual fnte volume of the Delaunay trangulaton, prevously defned, s called Vorono cell or Thessen polygon [42]. Each Vorono cell e assocated to node P conssts of the ponts Q such that d(q,p ) 6 d(q,p j ) for j =1,...,N and j (d(q,p ) s the dstance between Q and P ). The vertces of the Vorono cells are the crcumcentres of the Delaunay trangulaton. The storage capacty s assumed concentrated n the cells (nodes) n the measure of 1/3 of the area of all the trangles sharng each node. A lnear varaton of the pezometrc head H nsde each trangle of the mesh s assumed on the base of the values at ts three nodes. After ntegraton n space, the dfferental form of the predcton system (10) s: A dh dt þ X j wth Fl out ;j ¼ X m Fl out ;j ¼ K k ;j h5=3 ; A ¼ 1 3 Fl n ;m þ A p ; ¼ 1;...; N ð14aþ X n¼1;n T jk T j n d ;n ð14bþ where A s the area of cell, N T s the total number of trangles, jk T j n s the area of trangle n, d,n s the Kronecker delta equal to one or zero accordng f node s or s not a vertex of trangle n; Fl out ;j s the flux gong from cell to the any neghbourng downstream (n the potental scale) cell j wth H k j 6 H k, flux coeffcent Kk ; j wll be further defned, Fl n ;m s the flux enterng n cell from any neghbourng upstream cell m wth H k 6 H k m and p s source term n node. Soluton of system (14a) can be dsentangled n the sequental soluton of N equatons by approxmatng the r.h.s. wth ts mean value along the gven tme step, that s by settng: A dh dt þ X j Fl out ;j ¼ X m Fl n ;m þ A p ð15þ where Fl n ;m s the mean n tme value of the flux enterng from cell m, prevously solved, and p s the p mean value. At each tme step, the computatonal cells are ordered accordng to the decreasng value of ther potental (the pezometrc head), computed at the end of the prevous tme step and then are sequentally solved throughout the computatonal doman. After soluton of each ODE (15), the mean n tme total flux gong from cell to the neghbourng downstream cells can be computed by the local mass balance for cell, that s Fl out ¼ Fl n secondary element h kþ1=2 h k A þ A p Dt k T Fg. 2. The basc mesh and the dual fnte volume mesh. ð16þ where Fl out and Fl n are respectvely the total mean leavng and enterng fluxes, wth

7 1432 C. Arcò et al. / Advances n Water Resources 34 (2011) Fl n ¼ X m Fl n ;m ð17þ and h kþ1=2 s the fnal value of the water depth computed by the predcton step. The mean flux Fl out ;j gong from cell to cell j wth H k > H k j can be estmated by parttonng Fl out accordng to the rato between the flux Fl out ;j and the sum of the leavng fluxes at the end of the predcton step, that s: Fl out ;j ¼ Fl out Fl out ;j Pl Fl out ;l kþ1=2 kþ1=2 ¼ Fl out K k ;j P l Kk ;l ð18aþ where the sum s extended to all the neghbourng cells l wth H k l 6 H k. Fnally, the mean n tme fluxes enterng n cells j wth lower total head can be computed as: Fl out ;j ¼ Fl n j; ð18bþ After soluton of the ODE correspondng to cell, the next equaton to be solved s relatve to the cell j wth the maxmum pezometrc head among the unsolved ones and H k j 6 H k. Observe that, because of the chosen equatons sortng, the mean enterng fluxes wll be always known before each ODE soluton. Eq. (16) represents the local mass contnuty equaton ntegrated n space and tme nsde the Vorono cell and ts applcaton guarantees the global conservaton of the mass (see also Appendx A). The soluton of the predcton problem can be classfed as explct, because t depends only on the ntal state n the cell and on the nformaton (.e. the flux) comng from the upstream (n the potental scale) cells, prevously solved. Dfferently from the prevous MAST formulatons ([6,8]), we compute the flux coeffcent K k ;j (see Eq. (14b)) as: H k K k ;j ¼ c1 ;j Ek 1 þ c2 ;j Ek H k j 2 d j wth ð19aþ c 1 ;j ¼ d1 ;j ; c2 ;j ¼ d2 ;j f d 1 ;j P 0 and d2 ;j P 0 ð19bþ c 1 ;j ¼ d1 ;j þ d2 ;j ; c2 ;j ¼ 0 f d1 ;j P d2 ;j and d 2 ;j < 0 ð19cþ c 1 ;j ¼ 0; c2 ;j ¼ d1 ;j þ d2 ;j f d 2 ;j P d1 ;j and d 1 ;j < 0 ð19dþ where d j s the dstance between nodes and j and d m ;j s the dstance between the crcumcentre of each element m = 1, 2 sharng edge j from the same edge, that s: d m ;j ¼ ðx m x 12 Þðy j y Þ ðy m y 12 Þðx j x Þ qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff dðmþ ð19eþ ðx j x Þ 2 þðy j y Þ 2 where x m, y m are the crcumcentre coordnates, x 12 and y 12 are the coordnates of the common edge mdpont (pont P 12 n Fg. 3) Fg. 3. Crcumcentres P 1 and P 2 of elements 1 and 2 sharng edge j. and d(m)= 1 or 1 f drecton j s respectvely counterclockwse or not n trangular element m. Observe that d m ;j < 0 f the angle opposte to sde j n trangle m s obtuse. If the edge j belongs to the external boundary of the doman we set: K k ;j ¼ d1 ;j Ek 1 H k H k j d j ð20þ and we assume that d 1 ;j P 0 (as better explaned n the next secton). Coeffcent E k m s equal to: E k m ¼ 1 qffffffffffffffffffffffffff ð21þ jdh k m j n m where sub-ndex m marks all parameters of trangular element m. The new adopted space dscretzaton s smlar to the one of the standard lnear conformng Galerkn Fnte Element scheme. Accordng to Eq. (14b) and Eqs. (19a) (19e), the flux Fl out;m ;j movng from cell to cell j n each of the two trangles m sharng sde j, s computed as: Fl out;m ;j ¼ c m ;j Ek m H k H k j h 5=3 d j ð22aþ It can be shown that n the Galerkn formulaton the same flux s computed as [42,43]: Fl out;m ;j ¼ d m ;j Ek m H k H k j d j ^h5=3 m ð22bþ where ^h m s the average n space water depth nsde trangle m. In both cases Eqs. (22a) and (22b) approxmate the flux between cells and j, through sde d m ;j of the Vorono polygons of cell and cell j, due to the pezometrc head dfference H k H k j. The dfference of the MAST nter-cell flux formulaton wth respect to the Galerkn scheme, motvated and dscussed n the next secton for the case of Delaunay unstructured meshes, does not affect the flux computaton between two cells wth constant water depth and sharng parts of two acute trangles. Dffusve problem (11) and (12) s solved usng the same spatal dscretzaton adopted n the predcton problem, as well as a fully mplct tme dscretzaton. Integraton of Eq. (11) nsde each Vorono cell leads to the followng system: A Dt g þ X D k ðg ;j g j Þd X ;n ¼ D k ;j ð# j # Þd ;n ; ¼ 1;...; N n¼1;n T n¼1;n T wth D k ;j ¼ X m¼1;2 c m ;j Ek m d j ð23aþ ðh km l Þ 5=3 ð23bþ where d,n s the Kronecker delta, equal to one or zero accordng f node s or s not a vertex of trangle n and the sum n Eq. (23b) s extended to the two trangles m sharng sde j. l = f H k P H k j ; l ¼ j f Hk < H k j. Dfferently from the prevous formulatons of the dffuson coeffcents D k ;j, the new one provdes the same flux estmaton of the predcton problem, for gven element parameter E k m and node varable dfference, that s Hk H k j n the predcton problem and g g j n the correcton problem. Fully mplct tme dscretzaton provdes uncondtonal stablty, along wth some approxmaton error n the soluton [30]. The approxmaton error s small because ts magntude s of the same order of the computed correcton g, and the source term on the rhs of Eq. (23a) goes to zero along wth the tme step sze. Even durng abrupt potental changes, the potental correcton wll be small wth respect to the predcted change. Ths mples that the absolute error n the estmaton of the pezometrc correcton wll only

8 C. Arcò et al. / Advances n Water Resources 34 (2011) weakly affect the pezometrc fnal value computed at tme level k +1. The lnear system resultng from Eqs. (23) has order equal to the number of the nodes and s well condtoned, wth a matrx that s always symmetrc, postve defnte and, accordng to the new flux formulaton, strctly dagonally domnant, even n the case of Delaunay trangulatons wth obtuse trangles (see next secton). It can be shown (see Appendx B) that ths last property guarantees the steady-state monotoncty of the soluton, that remans regular also preservng the local mass conservaton, even n the parts of the doman wth sharp topographc changes, lke along the banks of a trapezodal channel. After soluton of the lnear system (23) s obtaned n the g unknowns, the pezometrc heads H at the end of the tme step are obtaned as: H kþ1 ¼ H kþ1=2 þ g ð24þ A major property of the MAST scheme, as heren formulated, s that n steady-state condton the correcton component vanshes for any arbtrary chosen tme step sze. Ths relevant result s due to the use of the same spatal dscretzaton for the computaton of both the convectve and the dffusve fluxes. The proposed scheme can be regarded as a fnte volume (FV) method where the control volume s the Vorono cell, smlar to the standard conformng Galerkn Fnte Element scheme. MAST scheme s a locally conservatve method, meanng that the sum of the fluxes over each cell edges equals the accumulaton term plus any sources or snks n the cell, and the flux s contnuous across each edge [29] (see Appendx C). 5. The requred generalzed Delaunay property If one edge j lnkng nodes and j s opposte to the obtuse angle of an element m, the dstance d m ;j of the crcumcentre from edge j, defned n Eq. (19e), s negatve and three possbltes exst. The frst possblty s that edge j s common to two elements and these have the sum of ther dstances s j ¼ d 1 ;j þ d2 ;j greater than or equal to zero, such that the crcumcentre P 1 of the obtuse trangle s located on the axs of the edge j between the same edge and the crcumcentre P 2 (Fg. 4a). It can be shown [20,42] that ths condton s equvalent to have the thrd node of the frst (or second) trangle out of the crcle defned by the three nodes of the second (or frst) trangle. Ths mples that the Delaunay property guarantees the condton (see Fg. 4b): d 1 ;j þ d2 ;j P 0 ð25aþ for all the edges of the mesh shared by two trangles. Most of the today avalable mesh-generators satsfy the Delaunay property, even f some exceptons occur around nternal boundares, or when the mesh densty s forced to change n gven subdomans. If the Delaunay property s satsfed, both the nter-cell fluxes computed by Eq. (22a) n the two elements sharng edge j are ether orented accordng to the dfference between the water levels of the two cells or zero. Observe that n the standard Galerkn Fnte Element dscretzaton, f the two element fluxes on the r.h.s. of Eq. (2c) are computed wth dfferent parameters E k m, the sgn of the total flux from node to node j can loose consstency wth the pezometrc dfference even f the mesh satsfes the Delaunay property and the sum of the dstances s j ¼ d 1 ;j þ d2 ;j s postve. The second possblty (Fg. 4b) s that the Delaunay property s not satsfed. In ths case t s stll possble to obtan a new mesh that satsfes condton (25a) for all the nternal edges, startng from the orgnal one, wthout changng the orgnal node locaton. Ths can be done by a seres of local edge swaps, where two elements sharng the same edge are changed n a new couple, sharng the same nodes but a dfferent edge, connectng the two nodes opposte to the prevous edge. See for example the new trangles obtaned n Fg. 5b by the orgnal ones of Fg. 5a. It can be shown [20] that the common edge satsfes the Delaunay property n at least one of the two confguratons. By teratng the same control for all the edges, the Delaunay property s quckly attaned for all the edges of the mesh that are shared by two trangles. The thrd possblty s that the element m s a boundary element and j s a boundary edge opposte to an obtuse angle. In ths case the flux coeffcent K k ;j n Eq. (20) remans negatve, even f the mesh satsfes the Delaunay property, because the dstance of the crcumcentre from the boundary edge s negatve. We defne Generalzed Delaunay mesh a Delaunay mesh where the condton: d 1 ;j P 0 Fg. 4b. Elements 1 and 2 do not satsfy the Delaunay property. holds for all the boundary edges. ð25bþ l c 1 m=1 c 2 m=2 j k Fg. 4a. Elements 1 and 2 satsfy the Delaunay property. Fg. 5a. Orgnal not Delaunay trangulaton.

9 1434 C. Arcò et al. / Advances n Water Resources 34 (2011) l 7 Swaped edge m=1 k c 1 c 2 m=2 j Fg. 6c. Generalzed Delaunay trangulaton after applyng the swap technque. Fg. 5b. Delaunay trangulaton after the mesh correcton (edge swap). If condton (25b) does not hold for one or more boundary edges, and/or common edges are fxed as nternal boundares, t s stll possble to obtan a Generalzed Delaunay mesh, also savng the nternal boundares. To ths am the two orgnal trangles sharng the nternal boundary or the orgnal trangle wth a boundary edge opposte to an obtuse angle are dvded n new trangles by addng new nodes along the orgnal edge. The new trangles have the same heght as the orgnal ones wth respect to the boundary edge, but the base length wll be a fracton of the orgnal one. After ths change, the same edge swap teratve procedure can be appled to the resultng edges, wth no excepton for the new ones located on the nternal boundary. It can be easly shown (see Appendx D) that the resultng mesh wll satsfy the Generalzed Delaunay property, f the number of new nodes s large enough. See an example n Fgs. 6a and 6b where the boundary edges 2 3 and 1 4 do not satsfy the Generalzed Delaunay property (Fg. 6a). The mesh s frst changed n a new mesh by settng a new node n both edges 2 3 and 1 4 (Fg. 6b), and then changed n a Generalzed Delaunay mesh by applyng the swap technque to the edge 4 5, that s changed wth the new edge 3 7 (Fg. 6c). The same spatal dscretzaton adopted for the convectve fluxes s also appled for the estmaton of the dffusve fluxes. In 2 2 New node 1 5 Fg. 6a. Orgnal trangulaton b/2 7 b New node b/2 Fg. 6b. New trangulaton after addng two new nodes. 4 Appendx E t s shown that ths mples for the resultng lnear system matrx the so called M-property [59], that s the negatvty of all the extra-dagonal matrx coeffcents. The M-property guarantees nter-cell fluxes wth a sgn that s always consstent wth the sgn of the correspondng water level dfference. An mportant consequence s the monotoncty of the steady-state soluton, when source terms are mssng, as well as the lack of spatal oscllatons [59]. Observe that the dffusve fluxes computed wth the spatal dscretzaton of Eq. (11) accordng to the standard Galerkn approach, are proportonal to a parameter T assumed constant nsde each element m and equal to [42,43]: T m;k ¼ E k m^h 5=3 ^h5=3 m ¼ m r n m DH k m ffffffffffffffffffffffffff ð26aþ The Galerkn approach guarantees the postve defnte condton (all the egenvalues greater than zero) of the fnal lnear system matrx, even f the Generalzed Delaunay condton does not hold, but does not guarantee the M-property [30,42]. On the other hand, f a non Generalzed Delaunay mesh s used wth the proposed algorthm, the teratve methods used for the soluton of the lnear system n the correcton problem can fal, because of the negatve egenvalues. Ths restrcts the use of the proposed algorthm to trangular meshes that satsfy condtons (25a) or (25b) n all the edges. Solvng the convectve and the dffusve problem usng the same computatonal cells and flux spatal dscretzaton s very mportant, because the use of dfferent formulaton for the computaton of convectve and dffusve fluxes can lead to small oscllatons n space and n tme even n the case of steady-state flow, when the dsspatve correcton s expected to go to zero n the MAST procedure. In the proposed algorthm, the fluxes computed by Eq. (26a) are proportonal to a parameter T gven by: T m;k ;j ¼ E k m h5=3 ¼ n m h 5=3 rffffffffffffffffffffffffff DH k m ð26bþ that cannot be thought as an element parameter. Ths mples that the velocty s not constant nsde the element and dvergence s not zero n each pont of the doman, even f local and global mass conservaton are both satsfed n the spatally dscretzed doman. 6. A sem-analytcal soluton of the predcton problem In the prevous formulaton of the MAST algorthm the soluton of each ODE problem (15) has been sought after numercally, adoptng a Runge Kutta scheme wth a self adaptng tme step, a fracton of the orgnal one [6]. In the present formulaton, an approxmated analytcal soluton s provded for the soluton of the problem (15). Call h k the water depth at the begnnng of the

10 C. Arcò et al. / Advances n Water Resources 34 (2011) tme step and h kf ts asymptotc steady-state value (.e. when dh / dt = 0), computed accordng to Eqs. (15) and (17), that s: 0 1 h kf wth Fl 0n ¼ P Fl 0n j K k ;j C A 3=5 ð27aþ ¼ Fl n þ A p ð27bþ Eq. (15) can be wrtten n dmensonless form as: dn ds ¼ 1 n5=3 ; n ¼ h ; s ¼ h kf dn ds ¼ n5=3 f n 5=3 ; n ¼ h ; s ¼ h k dt Fl0n A h kf dt Fl0n A h k f h kf > h k ð28aþ! h k 5=3 f h kf h kf < h k ð28bþ A seres soluton of Eqs. (28a) and (28b) s possble, but a good approxmaton can also be found wth a smaller computatonal tme by settng: Fg. 7. Functon c 3. n ¼ expðc 1sÞþc 2 expðc 1 sþþc 3 f h kf > h k ð29aþ n ¼ 1 þðn f 1Þ expðc 1sÞþc 2 expðc 1 sþþc 3 f h kf < h k ð29bþ wth a proper choce of the c 1, c 2 and c 3 coeffcents. Usng any c 3 value t s possble to match the ntal value n 0 and ts frst dervatve n 0 0 by settng: c 2 ¼ n 0 ð1 þ c 3 Þ 1; c 1 ¼ n 0 ð1 þ c 3 Þ 2 0 ðc 3 c 2 Þ c 2 ¼ 1; c 1 ¼ n 0 ð1 þ c 3 Þ 0 ðn f 1Þ f h kf > h k ð30aþ f h kf < h k ð30bþ Observe that functons (29a) and (29b) always match the ntal and the asymptotc values of the real soluton and guarantee ts tme second order convergence for any gven c 3 value. The c 3 coeffcent affects the maxmum error that s obtaned accordng to functons (29a) and (29b) usng dfferent tme step szes. Ths optmum depends on n 0 for case (a) and on n f for case (b). The optmum coeffcents have been computed numercally for dfferent possble n 0 and n f values by comparng functons (29a) and (29b) wth a numercal soluton computed usng a very small tme step. See n Table 1 and n Fg. 7 the computed optmum c 3 values. See n Fgs. 8a and 8b the numercal soluton of Eqs. (28a) and (28b) n the case of respectvely n 0 = 0 and n f = 0, compared wth the sem-analytcal solutons (29a) and (29b) correspondng to the optmal c 3 values (respectvely and ). The maxmum error computed wth the ntal condtons n 0 = 0, for h kf > h k,orn f = 0, for h kf < h k, s the worse one and t s smaller Table 1 c 3 coeffcent that provdes the smallest error for any possble s value. n, n f h kf > h k h kf < h k c 3 c Fg. 8a. Comparson of the soluton obtaned by the sem-analytcal method and the exact soluton: case n 0 ¼ 0 h kf > h k. Fg. 8b. Comparson of the soluton obtaned by the sem-analytcal method and the exact soluton: case n f ¼ 0 h kf < h k. than See also, n the same fgures, the sem-analytcal solutons correspondng to c 3 = 0 and c 3 = 1. These two solutons are equvalent to the analytcal closed form soluton of Eqs. (28) when the power exponent s approxmated respectvely to 1 or 2. Observe that no specal addtonal treatment for wettng and dryng procedure s requred n the proposed algorthm. An mportant feature of Eq. (15), n facts, s that t can be solved also wth zero ntal water depth value. Unless the spatal water level gradent at tme level k s close to zero around the cell, the knematc wave of the predcton problem propagates beyond the dry cell and ths allows the use of Courant number greater than one.

11 1436 C. Arcò et al. / Advances n Water Resources 34 (2011) Observe that the sem-analytcal soluton also exsts n the case n 0 ¼ 0 ðh kf > h k Þ. If, after the soluton of the predcton step, h km ¼ 0, the correspondng extra-dagonal terms of the system matrx are zero as well and zero fluxes are computed n all the elements surroundng cell. 7. Local head losses due to vertcal walls Dffusve models are fully sutable for the reconstructon of water level profles and of the vertcally averaged veloctes n most of the computatonal doman, for the smulaton of natural floods. In some cases and for some specal purposes, where vortcty or vertcal velocty components have to be estmated, dffusve models are no more adequate, but n other cases t s possble to account for the effect of downstream local energy losses by means of a fcttous change of the Mannng coeffcent. One of these cases s the restrcton of a rver cross secton gven by vertcal or subvertcal walls. Modellng a rver stream flow wth a dffusve 2D model s equvalent to compute the total dscharge as the ntegral of the dscharge per unt wdth, computed along the drecton transverse to the flow wth a vertcally averaged velocty that s dfferent on each element crossed by the vertcal secton. The same methodology s appled by most of the 1D models usng the so called Dvded Channel Method (DCM), along wth makng the further assumpton of a sngle energy slope nsde the secton [51]. The nconvenent of ths approach s that t underestmates the flow resstance n the case of vertcal or subvertcal walls, because the velocty reducton occurrng near the wall s not accounted for by the model. The reducton of conveyance capacty, caused by walls, can be restored by gvng to the elements nsde the restrcton an equvalent roughness coeffcent n p, dfferent from the natural one. n p s the roughness coeffcent the provdes, n steady-state flow condton, the same dscharge per unt wdth q p calculated takng nto account the shear stress nduced by the walls. The effectve conveyance capacty s estmated at the begnnng of the tme marchng procedure, applyng the Interactng Dvded Channel Method (IDCM) proposed by Huthoff et al. [25]. The IDCM s based on a parameterzaton of the nterface stress between adjacent flow compartments; assumng steady-state and 1D flow the authors have proposed the followng equatons to compute the flow velocty per unt bed slope n the compartments (see Fg. 9): where a s a dmensonless nterface coeffcent emprcally set equal to 0.02 [25], n j s the natural Mannng s roughness coeffcent and R j s the compartment hydraulc radus. Observe that, f the nterface coeffcent s set equal to 0, namely the lateral momentum transfer s neglected, Eq. (31) s equvalent to the standard DCM for a large enough number of compartments. Ths s because, f the compartment wdth s small enough, also the velocty and the dscharge per unt wdth n the two lateral compartments become neglgble, but ths affects the velocty dstrbuton nsde the entre secton only f a 0. In Fg. 10 we can see how the mean hydraulc radus, defned as: R ¼ 3=2 q p r ff ð34þ changes for normal flow nsde a rectangular secton accordng to the choce of the a coeffcent. The thrd curve of the same graph shows the hydraulc radus computed as the smple rato between the total cross secton area and ts wetted permeter. We can observe that ths value s ntermedate between those estmated assumng a compartment segmentaton and a = 0 or a = Before startng the marchng n tme smulaton, the dscharge per unt slope versus pezometrc level n the pth restrcton s computed as: Q p ¼ X j b j V j h j ð35þ At the begnnng of each tme step k, the equvalent roughness coeffcent n p s be computed as: n p ¼ ðhk Þ 5=3 B p Q p ðh k Þ ð36þ where B p s the restrcton wdth. Insde cross secton restrctons, lke between the ples of a brdge, a major head loss can also be gven by the exstence of an hydraulc jump. Once agan, t s possble to take nto account the jump effects by an artfcal ncrease of the natural Mannng coeffcent, after the jump exstence s checked out at each tme level. The jump exstence s tested by neglectng the transton effects and comparng the unform flow energy upstream the restrcton qga j ¼ qf j P j V 2 j þ h j 1=2 s j 1=2 þ h jþ1=2 s jþ1=2 ð31þ where q s the densty, g s the gravty acceleraton, A j s the compartment area, h j 1/2 refers to the nterface on the left and h j+1/2 to the nterface on the rght of compartment j. The correspondng shear stress and the dmensonless bed roughness are s jþ1=2 ¼ 1 2 aqðv 2 jþ1 V 2 j Þ f j ¼ gn2 j R 1=3 j ð32þ ð33þ Fg. 9. Subdvson of the cross-secton and notatons. Fg. 10. Dmensonless hydraulc radus.

12 C. Arcò et al. / Advances n Water Resources 34 (2011) wth the mnmum possble energy nsde the restrcton, for equal total dscharge. The upstream unform flow energy E 0 s equal to: E 0 ¼ n r r q 3=5 pff p þ ðr! r q p Þ 4=3 3=5 ð37þ n 2 ð2gþ 5=3 where n s the natural Mannng roughness coeffcent, r r s the contracton coeffcent gven by the rato between the total wdth nsde and upstream the secton restrcton and q p s the dscharge per unt wdth computed at begnnng of tme level k n a gven element nsde the restrcton. The mnmum crtcal energy s equal to: E k ¼ 3 2! q 2 1=3 p ð38þ g Observe n Fg. 11 the comparson between the unform flow and the crtcal state energy, as functon of parameters r r and. The unform flow energy s smaller than the crtcal state energy, wth subsequent hydraulc jump exstence, only wthn a restrcted slope range. Smlarly, one can see by comparng the q p exponents on the rhs of Eqs. (37) and (38) that unform flow energy can be greater than crtcal state energy for very small or very large dscharge values. Ths mples that temporal dscontnuty of the profle may appear upstream the restrcton durng very slow ncrement of the dscharge q p n quas-steady flow condtons, even wthout a wave front spatal propagaton. If the unform flow energy nsde the restrcton s too small to allow the computed dscharge, a total head loss reducton wll occur before the restrcton, and the saved energy wll be dsspated wth an hydraulc jump. The flow depth h m n the secton mmedately upstream the restrcton can then be computed usng the followng ratng curve, for fxed mnmum E k energy calculated by Eq. (38): h 3 m E kh 2 m ¼ q2 m 2g ð39þ where q m s the upstream dscharge per unt wdth mmedately before the restrcton. The fcttous roughness coeffcent n r s computed n order to obtan a total head loss nsde the restrcton equal to the dfference between the mnmum upstream energy and the pezometrc level Hv mmedately downstream the restrcton. Ths can be done by ntegratng the followng dffusve momentum equaton: dh dx ¼ n2 r q2 p ðh zþ 10=3 to get: ð40þ Fg. 11. Dmensonless energy E 0 /(nq p ) 3/5 and E k /(nq p ) 3/5 versus parameters and r r. h m þ z m H v ¼ n 2 r Fg. 12. Restrcton scheme and notaton. Z x2 x 1 q2 p dx 10=3 ðh zþ ð41þ where x 1 and x 2 (see Fg. 12) are the ntal and fnal restrcton abscssa. The fcttous Mannng coeffcent n r s computed as the root of Eq. (41). If the restrcton wdth s constant, computaton can be smplfed by assumng a constant average bed elevaton z a nsde the ntegral, to get: " # n 2 r ¼ 7ðh 1 m þ z m H v Þ 1 3q 2 p ðh m z a Þ 1 ð42þ 7=3 ðh v z a Þ 7=3 The procedure has been compared wth lterature results. Tang et al. [53] compare the dscharge hydrographs computed at the end of a rectangular channel usng three dfferent types of 1D Muskngum Cunge models: two VPMC (Varable Parameters Muskngum Cunge) and one CPMC (Costant Parameters Muskngum Cunge) methods. The channel has 50 m wdth, 100 km total length, 0.025% bed slope and s/m 1/3 Mannng s coeffcent. The computatonal doman has been dscretzed usng a mesh wth rght-angle trangles and two parallel lnes of 2001 nodes, each one wth a dstance of 50 m from the next one. The flow depth correspondng to a constant dscharge of 100 m 3 /s has been used as ntal condton. The Drchlet boundary condton was appled to the two downstream fnal nodes. Even f the depth/wdth rato s smaller than 1:10, we can see n Fg. 13 that the routed dscharge hydrograph s strongly affected by the choce of the a coeffcent n Eq. (32), that s by the type of velocty dstrbuton assumed n the vertcal cross sectons. In the case of nterface coeffcent a equal to 0 (equvalent to usng the DCM method), the routed dscharge hydrograph computed by the proposed MAST-2D dffusve model s equal to the hydrograph obtaned by the DORA 1D dffusve model [38] adoptng the DCM method and has a peak value hgher than that computed by the Muskngum Cunge models (Fg. 13). Ths seems correct, because assumng a = 0 s equvalent to neglect the wall effects and underestmate the hydraulc resstance. If the lateral momentum transfer s not neglected (a = 0.02), the results show a greater peak reducton wth respect to the Muskngum Cunge model, that compute the hydraulc resstance as functon of the hydraulc radus of the overall secton. Ths s consstent wth the result already shown n Fg. 10, where dfferent ways of computng the average hydraulc radus n normal flow condtons have been compared.

13 1438 C. Arcò et al. / Advances n Water Resources 34 (2011) Fg. 13. Outflow dscharge hydrographs computed by the proposed and the Muskngum Cunge models. 8. Numercal experments In ths secton, we present fve numercal tests for the model valdaton n the most general 2D-case Test 1. Stablty wth regard to the Courant number In the frst test we nvestgate the stablty of the model results aganst the sze of the element CFL number, computed as: CFL ¼ V e Dt pffffffffff A e ð43þ where A e and V e are respectvely the element area and the velocty computed n the center of the element assumng a water depth equal to the average value at the element nodes. A symmetrc square [10,000 m 10,000 m] doman has been dscretzed wth the unstructured Delaunay computatonal mesh shown n Fg. 14, wth 1961 nodes and 3758 elements. In order to magnfy the rregularty of the mesh, three nternal boundares have been assgned to the doman. The open source mesh generator NETGEN [37,46] has been used to generate the ntal mesh. The algorthm presented n Secton 5 has been fnally used to obtan the fnal mesh and to guarantee the Generalzed Delaunay property, wthout changng the assgned nternal and external Inflow Inflow Inflow 8000 Inflow 6000 y 4000 Inflow x Inflow N Fg. 14. Test 1. Computatonal mesh.

14 C. Arcò et al. / Advances n Water Resources 34 (2011) boundares. The Mannng coeffcent s s/m 1/3, the bottom elevaton of pont (0,0) s equal to zero and the bottom slope s n both x and y drectons. A constant zero water depth has been assgned on the East and South external doman sdes as Drchlet boundary condtons. A symmetrc trangular nflow hydrograph n the boundary nflow nodes shown n Fg. 14, wth peak dscharge equal to 2500 m 3 /s and peak tme equal to 8 h, has been gven as Neumann upstream boundary condton at the North and West sde. Intal condton s h 0 =0. Fve smulatons wth dfferent tme step sze have been carred out on the same mesh. Table 2 shows the tme step sze Dt, the maxmum CFL value calculated accordng to Eq. (43) n the doman durng the entre smulaton, the mean CPU tme per computatonal node and per teraton and the relatve error e of the soluton at the peak tme, between the 1st and mth smulaton, calculated as: vffffffffffffffffffffffffffffffffffffffffffffffffff uðh e m h ref Þ 2 m ¼ t ð44þ ðh ref Þ 2 To test the robustness of the model wth respect to the mesh rregularty, the soluton of the second smulaton (Dt = 40 s) at the peak tme t = 32,000 s (see Fg. 15) has been compared wth the results obtaned over an unstructured mldly dstorted mesh wth 1931 nodes and 3700 elements (see Fg. 16a). Fg. 16b shows the absolute values of the relatve dfference between water levels calculated over both the mldly and hghly unstructured meshes. The relatve dfference has been computed as: Dhðx; yþ ¼ hre h r ^h re ð45þ where h re and h r are the water depths computed wth respectvely the more and less regular mesh and ^h re s the average water depth value computed wth the regular mesh over all the doman. Observe that the relatve dfference s less than 1% over most of the computatonal doman and the larger dfferences are close to the where h m s the maxmum water depth computed n node durng the mth smulaton and h ref s the same water depth computed durng the 1st smulaton wth the mnmum tme step sze. Node s located n the center of the doman. Observe that the stablty and accuracy of the model s obtaned also for very large values of the CFL number. In ths test case, f CFL s less than 21.5, the maxmum relatve error e s less than 1%. Moreover, the ncrease n the CFL number does not lead to any ncrease n the mean CPU tme. If the CFL number s greater than 70 the stablty of the model s guaranteed, but n the transent soluton there s a numercal dffuson, leadng to a flattenng and a deformaton of the water depth curve (Fg. 15). Table 2 Test 1. max. CFL numbers, errors and mean CPU tmes. Smulaton Dt [s] Max CFL Mean CPU tmes [s] e E E E E E E E E E 02 Fg. 16a. Test 1. Mldly dstorted unstructured mesh. Fg. 15. Test 1. Water depth curve n the center of the doman.

15 1440 C. Arcò et al. / Advances n Water Resources 34 (2011) connecton accordng to the same procedure descrbed n Secton 5. Moreover, the same boundary and ntal condtons as n test 1 have been appled. For the frst smulaton (coarsest mesh), a tme step of 150 s has been used, wth a maxmum CFL number equal to In order to obtan a CFL number smlar n each refned mesh, the tme step has been halved at each refnement. The CPU tme of the total smulaton, of the convectve step and of the dffusve step, has been recorded for each mesh. Table 3 shows the specfc CPU tmes of a sngle processor Intel Q GHz, per node and per tme step. The trend of the CPU tme versus the number of nodes s dfferent for the predcton and correcton steps. The mean specfc CPU tme for the convectve step remans almost constant, wth a growth due only to the cells sortng. The specfc CPU tme for the soluton of the lnear system assocated to the dffusve problem grows very slowly, and the total CPU tme s only a bt more than proportonal to the number of nodes. The growth rate b, measured as the exponent of the relatonshp: CP ¼ N b ð46þ where N s the number of nodes and CP s the average CPU tme per each tme step, s only 1.10 n the proposed test (see Fg. 18). Fg. 16b. Test 1. Absolute value of the relatve dfference between water surface calculated on the mldly and hghly dstorted mesh. Fg. 17. Test 2. Refnement scheme Test 3. The convergence rate The thrd numercal test s focused on the order of convergence of the proposed model. Because test cases to be used for comparson between analytcal and numercal solutons n 2D transent condtons are not avalable, a dfferent procedure has been appled. The followng arbtrary analytcal soluton H = H(x, y, t) s gven for Eq. (4), where the source term on ts rhs s computed by tme and space dfferentaton of the known H on the lhs of the same Eq. (4), pffffffffffffffffffffffffffffff H ¼ 0:001 x 2 þ y pffffff 2 2 þ h 2 h pffffffffffffffffffffffffffffff tanh k ut x 2 þ y 2 þ 1 ð47þ where x and y are the doman pont coordnates, t s the tme and k, u, h are constants equal to , 1 and 3 respectvely. The analytcal soluton has been assgned on the same [10,000 m 10,000 m] boundares, where the soluton s strongly affected by the element sze, but the same dfference s very small around the nternal boundares, where the mesh s more dstorted Test 2. Computatonal performance nvestgaton In the second numercal experment, the computatonal performance of the proposed model has been nvestgated, usng the same mesh of the prevous experment (see Fg. 14). The orgnal mesh has been refned by dvdng each element n four equal trangles and connectng the mdponts of the three sdes of each trangle. See n Fg. 17 the refnement scheme. Three refnement levels have been carred out. After each refnement, the Generalzed Delaunay condton was guaranteed by changng the nodes Fg. 18. Test 2. Trend of the CPU tmes n Eq. (46) requred for the soluton of the convectve and dffusve steps. Table 3 Test 2. Mean CPU tmes. Refnement Number of elements Number of nodes Mean CPU tmes (convectve) [s] Mean CPU tmes (dffusve) [s] E E , E E , E E , E E 06

16 C. Arcò et al. / Advances n Water Resources 34 (2011) Table 4 Test 3. Convergence rates. Refnement r c upstream node r c front wave node r c downstream node has been halved at each refnement n order to mantan almost constant the CFL number from one refnement to the other. The relatve error on each node and for each mesh has been estmated by Eq. (49a) at the smulaton tme t = 7000 s: sffffffffffffffffffffffffffffffffffffffffff ðh e n h a m ¼ Þ2 ð49aþ ðh a Þ2 Fg. 19. Test 3. Plot of the exact soluton for the pezometrc head of Eq. (47) at t = 7000 s. doman used n the prevous tests. The Mannng coeffcent s s/m 1/3 and the followng bottom elevaton z s gven: z ¼ 0:001 x p þ ffffff y 2 ð48þ Functon H n Eq. (47) s shown n Fg. 19 at tme t = 7000 s. Gven the soluton H, the source term p n Eq. (4) s computed n each pont and at any tme by solvng the lhs of the same equaton. Observe that the proposed functon H has zero flux along the South and West sdes of the doman and t s assgned as Drchlet boundary condton n the two other sdes. The same mesh refnement as n test 2 has been used. The tme step Dt s 100 s for the coarsest mesh, correspondng to a maxmum spatally averaged CFL number equal to The tme step where h n and h a are respectvely the numercal and the analytcal water depth computed n node. The rate of convergence s defned by comparng the relatve errors of two consecutve mesh levels. Assumng the relatve error obtaned for mesh level m proportonal to a power of the lnear sze of the area of the mean trangle n the mesh, that s: pffffffffffff rc e m ¼ ð49bþ A m where A m s the area of the mean trangles n the mesh refnement p level m and ffffffffffff A m represents a measure of ts lnear sze, the rate of convergence r c s computed by comparng the relatve errors of two successve refnement levels m and m + 1: e log m e mþ1 r c ¼ ð50þ log 2 Table 4 shows the rates of the convergence calculated at three nodes located respectvely on the front, upstream and downstream the advancng wave (see Fg. 20). The results show that: (a) the rate of convergence ncreases along wth the mesh densty and ths guarantees robustness n the case of very coarse meshes, (b) the asymptotc rate of convergence s hgher n the areas wth larger pezometrc gradents, even greater than 2, but smaller around Fg. 20. Test 3. Selected nodes for the convergence rate calculaton.

17 1442 C. Arcò et al. / Advances n Water Resources 34 (2011) flatter areas. Ths can be explaned by the proportonalty of the fluxes to the root of the norm of the pezometrc gradent, that leads to an nfnte senstvty of the fluxes wth respect to the pezometrc gradent around ts zero value Test 4. Comparson wth wettng dryng lterature tests Gourgue et al. [21] have valdated the WD opton of ther fully dynamc model usng a test already proposed by Balzano [9] n one-dmensonal form. They proposed a flux lmtng WD approach for a DG FE method for the soluton of the fully dynamc SW equatons. A basn wth unform bottom slope, length m n x drecton and wdth 7200 m n y drecton s dscretzed by an unstructured trangular mesh. The adopted mesh (Fg. 21) has a node densty smlar to the mesh n [21], wth 690 elements and 383 nodes. Mannng coeffcent s 0.02 s/m 1/3, bottom elevaton s constant n y drecton whle the slope n x drecton s The doman has upstream zero flux boundary, downstream open boundary and the ntal condton s pezometrc head equal to zero. At the open downstream boundary, a snusodal water level varaton s assgned. The ampltude and the perod of the oscllaton are respectvely 2 m and 12 h. In ths case the results of the dffusve model s smlar to the results of the complete model, because the tme perod of the downstream boundary condton s large enough wth respect to the travel tme of the generated wave, needed to cover the doman length. Observe n Fg. 22 the free surface shape computed by both models every 20 mn. The wet-dry lmt s very smlar at all the nvestgated tmes and negatve depths, as well as artfcal spatal oscllatons are mssng n both models. The pecularty of our model s that no specal treatment s needed for the elements located near the wet-dry lmt. The proposed numercal model has been tested also for the thrd Balzano test. The basn contans a small reservor and the elevaton of the bottom s calculated by the followng analytcal expressons: z ¼ x=2760 f x m or x P 6000 m z ¼ x= =23 f 3600 m < x m z ¼ x=920 þ 100=23 f 4800 m < x < 6000 m ð51þ Fg. 21. Test 4. Frst Balzano test. Computatonal mesh. The ntal condton s a pezometrc head equal to 2 m and the boundary condton s a snusodal decay appled at the open boundary. At the open boundary the water depth decreases from 7 m to 3 m wthn 6 h (half the snusodal perod). After ths perod, the water depth at the open boundary s left ndefntely at 3 m n order to test whether water s leakng through the dry area. The computatonal mesh s composed of 518 elements and 296 nodes (see Fg. 23). The surface n the reservor should asymptotcally reach an horzontal plane at the level of the local peak of the bathymetry n the pond. Many numercal models fal n representng the water surface profle because water fluxes do not vansh as Fg. 22. Test 4. Frst Balzano test. Water surface every 20 mn (thn lnes) and bottom channel (thck lne). Results of the MAST model (a) and (c); Gourgue et al. [21] model results (b) and (d).

18 C. Arcò et al. / Advances n Water Resources 34 (2011) Fg. 23. Test 4. Thrd Balzano test. Computatonal mesh. long as the pressure gradent term operates, even when the mean surface level nsde the reservor s below the local peak of the bathymetry and the pond dres up. When water depth s zero at x = 4800 m (the rght end sde of the pond), numercal fluxes computed n the proposed model accordng to Eq. (22a) vansh. In Fg. 24, the computed water surface profles are shown and compared wth the ones computed by [21] after 100 h. Observe that the expected fnal water level s perfectly smulated n the reservor. The reference results have been computed usng a smlar unstructured mesh (see [21]) Test 5. Comparson wth a complete model usng an overflow test case In ths last numercal test the results of the dffusve model have been compared wth the results of a fully dynamc model. In ths test case both models could be reasonably appled for the water level and velocty computaton. The complete model appled here for comparson s the Godunov-type fnte-volume model of BreZo [12,13], that solves the depth-averaged Shallow Water Equatons on an unstructured grd of trangular cells. The algorthm of BreZo uses Roe s approxmate Remann solver to compute fluxes, a multdmensonal lmter for second-order spatal accuracy and predctor corrector tme steppng for second-order temporal accuracy. The model features a specal technque for the treatment of the partally wetted cells based on equatons that defne exactly the relatonshp between free surface elevaton and water content of each cell (Volume/ Free-surface Relatonshps, VFRs). These equatons are appled at each tme step to consstently track flud volume and the free surface elevaton (whch s mportant for flux evaluaton) n partally submerged cells. As for the soluton algorthm, n the predctor step the soluton s updated by solvng the SW equatons n terms of the prmtve varables, to mprove the computatonal effcency, whle n the corrector step the conservatve, ntegral form of the SW equatons s solved. Whereas the contnuty equaton s updated n all cells, the momentum equatons are not solved n partally wetted cells (.e., where at least one of the three nodes s not submerged), where the velocty s set to zero n order to avod spurous acceleratons. The test case s the steady-state reconstructon of a rver floodng. The spatal doman s gven by a trapezodal channel crossng a flat area wth a channel depth reducton n the central part of the doman (see n Fg. 25 the plan vew and the cross secton). A frst unstructured mesh adopted for spatal dscretzaton has nodes and elements. In ths mesh, only few nodes were used to represent each channel secton morphology and zero nodes were located nsde the area of the rver banks. The Fg. 24. Test 4. Thrd Balzano test. Bottom channel (thck lne) and poston of the water surface at ntal tme and at equlbrum (thn lnes). Results of the MAST model (a); Gourgue et al. [21] model results (b). 50 m (a) Plan (b) Cross secton = = = m 20 m 8 m = 0.01 t 1 1 Fg. 25. Test 5. Compound channel. steady-state dscharge s 9 m 3 /s, dstrbuted as Neumann boundary condton along the central part of the upstream boundary sde (from y =21mtoy = 29 m); the Mannng coeffcent s 0.03 s/m 1/ 3 and the Drchlet water depth, at the downstream boundary, s equal to normal flow depth correspondng to the upstream condton. Smulaton tme s 10,000 s, large enough to get steady-state flow n the channel, startng from completely dry bed condton. The adopted Dt for the MAST smulatons s 5 s (2000 total tme

19 1444 C. Arcò et al. / Advances n Water Resources 34 (2011) teratons), whle BreZo scheme adapts the tme step sze accordng to the condton CFL = 0.9; Dt at steady-state condtons s 2.48 s. In the lower part of the channel, flow remans nsde the trapezodal secton, whle floodng occurs n the central and n the upper parts. The comparson of the solutons obtaned wth the proposed model and the complete BreZo model shows sgnfcant dfferences n the flooded areas (see Fg. 26). The computatonal mesh has been refned as prevously explaned. Tme step for MAST smulatons has been reduced to 2.5 s, whle the Dt at steady-state condtons for BreZo smulatons s n ths case s. After the refnement, the proposed model provdes a soluton very smlar to that obtaned wth the frst mesh (see Fg. 27a). On the other hand, the complete model gves n ths case a soluton dfferent from the soluton obtaned wth the frst mesh and smlar to the one computed by the MAST model usng the same refned mesh. Accordng to ths result, BreZo soluton underestmates the flooded area usng the coarse mesh (see Fg. 27b). A frst motvaton of ths dfference s that n BreZo the momentum equatons are not solved n partally wetted cells, whch adds a numercal frcton n these cells. Ths effect s notceable when coarse mesh are used, and quckly decreases as the computatonal grd s refned, as descrbed n [13]. A second motvaton can be found n the dfferent topography represented by the two meshes. Dfferent node elevatons correspond to dfferent source term and, for gven specfc energy, to dfferent water levels. The results of ths tests are consstent wth the observaton of Secton 2 and suggest that the stronger stablty of the dffusve model can lead to better results n the computaton of the water levels even when the approxmaton of the ground elevaton s due not to measurement error, but to the same spatal dscretzaton. Fg. 28 shows the velocty feld computed by the MAST scheme and Fg. 29 shows the scatters from the one computed by the BreZo model. Man dfferences occur n the flooded areas, eventhough, due to the dfferent scales adopted for the graphcs, these dfferences are very low. A comparson of the CPU tmes requred by the MAST and BreZo models for the soluton of the present test usng the fner mesh and a sngle processor Intel Q GHz has been done. Total computatonal tme requred by MAST scheme s s (13% for the soluton of the predcton step and 87% for the soluton of the correcton step). Computatonal tme of the BreZo model s approxmately 39 tmes hgher. The water depths n two nodes of the bank (see Fg. 30) computed usng the coarse mesh along the tme are shown n Fg. 31. Observe that numercal oscllatons never occur and the dffusve correcton g obtaned by the soluton of Eq. (11) quckly converges to zero Test 6. Applcaton to a real case: Ventur channel of Imera rver (Scly) The proposed model has been appled for the smulaton of the backwater profle caused by the Ventur channel located n the southern part of the Imera rver, n Scly. Fg. 32a shows the geometry of the structure. Fg. 32b shows the vertcal walls of the restrcton mmedately after a flood event and the maxmum level Fg. 26. Test 5. Steady-state water depth upstream the flat area. (a) MAST model and (b) BreZo model [12] (coarse mesh). Fg. 27. Test 5. Steady-state water depth upstream the flat area. (a) MAST model and (b) BreZo model [12] (refned mesh).

20 C. Arcò et al. / Advances n Water Resources 34 (2011) Fg. 31. Test 5. Computed water depth at the montored nodes. Fg. 28. Test 5. Detal of velocty feld n the refned mesh calculated by MAST model. Fg. 32a. Test 6. Geometry of the Ventur channel. Fg. 29. Test 5. Detal of the scatters between velocty felds calculated by MAST model and by BreZo model (refned mesh). Lower node x = 400 m Fg. 30. Test 5. Montored nodes poston. Fg. 32b. Test 6. Hstorcal water surface profle.

21 1446 C. Arcò et al. / Advances n Water Resources 34 (2011) Fg. 33. Test 6. Steady-state profles n the mddle of the channel. reached by the water surface durng the event s marked wth a red lne. In a prevous study [36] the Mannng s coeffcent has been estmated equal to 0.04 s/m 1/3 and the maxmum dscharge, assumed constant along all the Ventur channel, equal to 1600 m 3 /s. Normal flow depth has been assumed n the downstream secton. In order to calculate the backwater profle, the roughness coeffcent nsde the restrcton has been automatcally calculated usng the procedure descrbed n Secton 7. The water levels computed by the proposed model have been compared wth both the marked elevatons and the steady-state water surface profle calculated by the complete model of BreZo usng the same mesh. Fg. 33 shows the central vertcal cross secton of the water surface profle. In spte of the many parameter uncertantes and errors, we can observe n the upstream secton of the Ventur channel a good match between the results of the MAST model and the hstorcal data, due to the reconstructon of the local energy dsspaton carred out wth the procedure explaned n Secton 7. Results of the complete model are more close to the hstorcal data only nsde the Ventur channel, where the dffusve model s unable to reproduce the profle of the supercrtcal flow. All these propertes make the algorthm specally suted for ts mplementaton n the context of early warnng systems. Dsclosure statement All the Authors of the paper, Costanza Arcò, Marco Snagra, Lorenzo Begnudell and Tullo Tuccarell, dsclose any actual or potental conflct of nterest ncludng any fnancal, personal or other relatonshps wth other people or organzatons wthn three (3) years of begnnng the work submtted. The Authors declare also that have partcpated n the research and artcle preparaton and ther ndvdual contrbuton to the paper s equally dstrbuted. The Authors, Costanza Arcò, Marco Snagra, Lorenzo Begnudell and Tullo Tuccarell Palermo, 11/04/2011. Acknowledgements The research has been funded by the PRIN 2008 program Integraton of hydraulc measurements n rvers for dscharge and hydraulc resstance parameter montorng. Appendx A. Mass conservaton 9. Conclusons A new algorthm for the smulaton of parabolc 2D SW equatons n strongly unstructured meshes has been developed, startng from the numercal structure of the prevous DORA model [56]. The algorthm s amed manly to the smulaton of gradually varyng flows, wth contnuous head losses, but the backwater effect of two types of localzed head losses, manly the effect of vertcal walls and hydraulc jumps, can be easly ncluded n the model. The most mportant mert of the algorthm s the conservaton of mass, the tme step uncondtonal stablty, ts robustness wth respect to abrupt parameter changes, lke the topographc gradent, and, most mportant, the very low growth of the computatonal burden versus the number of nodes. The power law exponent of the computatonal burden s about 1.10, that s very close to the value of the explct algorthms. The results of the proposed model have been compared wth the results of a well-known complete model, wth a completely dfferent numercal structure. The results suggest that the use of dffusve models nstead of a complete one can lead not only to computatonal tme savng, but also to more accurate predcton, because of the smaller senstvty of the dffusve model to the nput topographc error wth respect to the senstvty of the complete model. Accordng to Eq. (16), t s possble to wrte Eq. (18a) as: DtFl out ;j ¼ Fl 0n Dt A h kþ1=2 h k K k ;j Dt P l Kk ;l where Fl 0n ða:1þ ¼ Fl n þ A p ða:2þ Summng all Eq. (A.1) for =1,...,N, one gets: X A h kþ1=2 h k ¼ Dt X Fl 0n Fl out ða:3þ Because for each lnked couple of nternal cells and j (see Eq. (14b)), Fl out ;j ¼ Fl n j; ða:4þ the rhs of Eq. (A.3) s equal to the dfference among the ncomng and leavng boundary fluxes, plus the sum of the source terms appled to the computatonal doman (f dfferent from zero), that s: X A where h kþ1=2 h k ¼ Dt Fl n bou Flout bou þ _ P ða:5þ